Proof a set is connected

Question

I have the set $$ S=\{(x,y)\in\mathbb{R}^2:(y^2-64)(x^2+y^2-16) = 0 \} $$ I want to show that it is connected and I wasn't sure how to start, the first idea I had was to show the only subsets of S which are open and closed are $\emptyset$ and $x$ as this is equivalent, but again I can't work out where to start even with sketching it out. I couldn't see anything obvious. Any ideas even for a starting point would be great?

Answer 1

That set is not connected. It consists of the union of

• the line $y=8$;
• the line $y=-8$;
• the circle centered at $(0,0)$ with radius $4$.

The line $y=8$ is a closed subset of $S$ (since it is a closed subset of $\Bbb R$), but it is also an open subset of $S$, since it is equal to $S\cap\{(x,y)\in\Bbb R^2\mid y>4\}$, and $\{(x,y)\in\Bbb R^2\mid y>4\}$ is an open subset of $\Bbb R^2$.